Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#11597 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | #102 |
| commutator | C8 | C22×C8×C127 |
| Frattini, cosocle | C4×C8 | C23×C127 |
| socle | C22×C127 | C2×#102 |
| prime radical | C2×#135×C127 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C2×C42×C127 ◁ C2×C4×C8×C127 ◁ G | |
394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 3 generators on 167 points:
(1,2,3,5,4,6,7,8)(9,10)(11,20,23,37)(12,16)(13,27)(14,19)(15,22,31,39)(17,25)(18,26)(21,36,24,33)(28,35,40,30)(29,32)(34,38)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(9,11)(10,15)(12,21)(13,28)(14,23)(16,30)(17,33)(18,35)(19,31)(20,32)(22,38)(24,26)(25,39)(27,37)(29,40)(34,36);(9,12,14,26)(10,16,19,18)(11,21,23,24)(13,25,29,38)(15,30,31,35)(17,32,34,27)(20,36,37,33)(22,28,39,40)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 4585);